When it comes to the development of mathematics in the 20th century, and even the development of mathematics in the 21st century, it is actually inseparable from the influence of Hilbert's 23 question, Hilbert 23 question can be said to be the highest program guiding the development of mathematics in the 20th century, so what is Hilbert 23 asking? Let's take a look at it today.
If it comes to the mathematician who had the greatest influence on mathematics in the 20th century, then Hilbert is the one who cannot be separated, and the reason why he has a profound influence on the mathematical community is not only because of his mathematical achievements, but also because he is the leader of the Göttingen School of Mathematics, which created the mathematical glory of the first thirty years of the 20th century, is the cradle in the hearts of countless mathematicians, and its aftermath still stirs the entire mathematical community today. Hilbert was thus hailed as the "uncrowned king" of mathematics.
David Hilbert (January 23, 1862 – February 14, 1943)
To put it simply, Hilbert was the master of the martial arts alliance in the mathematical field at that time, and he trained a large number of martial arts masters, and the entire martial arts talent was abundant.
Hilbert was born in The Former Soviet Union near Kaliningrad near Velaud, from an early age he showed a strong interest in mathematics, he even defied his father's objections, entered the University of Königsberg to study mathematics, here met his lifelong close friend Minkowski, the most rare thing in life is to meet a friend who is like-minded with you and works hard with you, Minkovsky is also a mathematical genius, he is Einstein's teacher, Minkowski space-time provides a framework for the establishment of general relativity.
Hilbert and Minkowski often walked together to discuss mathematical problems, the exchange of two mathematical geniuses collided with a fierce spark, almost all of Hilbert's important achievements would be reviewed by Minkowski in advance, the friendship between the two was good, Minkowski saw that there was no Hilbert speech at the mathematical conference, and he was so disappointed that he did not want to participate in the conference.
A few days before Minkowski's death, Hilbert also told Minkowski that the next time he would give proof of Walling's theorem, hoping that Minkowski would participate, and minkowski died remembering that he had not heard Hilbert's proof of Waring's theorem, and that he wanted to see Hilbert again before he died.
Hilbert's series of mathematical achievements are inseparable from the warm exchange with Minkowski and the collision of fireworks, Hilbert's life of research on a number of mathematical topics, including invariants theory, algebraic field theory, geometric foundations, integral equations, general mathematical foundations, interspersed with research topics such as: Dilicre principle and variational method, Walling problem, eigenvalue problem, "Hilbert space" and so on.
Speaking of Hilbert Space, there is another interesting thing, the mathematical terms named after Hilbert are so many that even Hilbert himself does not know. At one point, Hilbert asked his colleagues in the department, "What is Hilbert Space?" (Hilbert space is a generalization of Hilbert's euclid space, but the name is not obtained by him, mathematicians generally study some problems, the names are later people to help them obtain, such as Hilbert space is von Neumann in his 1929 book on the unbounded Ermith operator, the earliest use of the term "Hilbert space".) ）
It was because of Hilbert's great talent in mathematics that Klein, the leader of the Göttingen School, admired Hilbert and believed that the Göttingen School of Mathematics was in need of someone like Hilbert as a successor. I have to say that Klein's vision of people is really accurate, the ability to dig people is first-class, Hilbert, Minkowski These are all he dug up Göttingen, Clay because later Hilbert led the Göttingen School to glory to lay the foundation.
Hilbert, who led the Göttingen school of mathematics, attracted young people from all over the world to Göttingen like pilgrimages with his unparalleled charm, and he alone supervised seventy or eighty Doctorates. A large number of young scholars flocked to Göttingen, not only from Germany and Europe, but also from Asia, especially the United States. According to statistics, there were 114 American mathematicians who obtained foreign degrees between 1862 and 1934, of which 34 received doctorates in Göttingen, and many influential papers at that time were written in German. The Tingen School became the cradle and holy place of mathematicians in the world and was the international mathematical center.
At that time, the loudest slogan among mathematicians students around the world was "Pack your backpack and go to Göttingen".
Nearly half of the most famous mathematicians in mathematics at that time were from the Göttingen school of mathematics, and Gominkovsky provided a mathematical framework for special relativity— Minkowski's four-dimensional geometry; Weyl was the first to propose the theory of gauge fields and provided a theoretical basis for general relativity; von Neumann provided a rigorous mathematical foundation for newly born quantum mechanics and developed functional analysis; and "the mother of modern mathematics" Noether laid the foundation for abstract algebra with general idealism, and stimulated the development of algebraic topology on this basis Curang is a master of applied mathematics, and his work on solving partial differential equations has cleared the way for a range of practical topics such as aerodynamics.
"The Greatest Female Mathematician" – Amy Knott
At this time, the stars of Göttingen were shining brightly, and everyone was free to wander in the temple of mathematics, allowing the sparks of thought to collide.
The Göttingen School, the third Hilbert on the left, do you know anything else?
In addition to training a large number of talents for the mathematical community, Hilbert in 1900, which was the end of the 19th century and the arrival of a new century, coincided with the second International Congress of Mathematicians, when Hilbert was already the leader of the Göttingen School, and naturally the first choice for the keynote speech of the conference.
At first, Hilbert was thinking about whether to defend pure mathematics, which is also called basic mathematics, which is a special study of mathematics itself, not for practical application, to study the internal relationship of mathematical laws abstracted from the objective world, and it can also be said to be the study of the laws of mathematics itself.
But then Hilbert changed his mind, thinking that it was time to summarize the development of mathematics since the 19th century, discuss the new direction of mathematics in the 20th century, and propose some important problems that mathematicians should concentrate on solving. Hilbert's idea was enthusiastically supported by Minkowski, who believed that the most attractive subject matter was to look forward to the future of mathematics, so Hilbert chose it.
So Hilbert spent six months preparing his speeches from January 1900 to July, and then spent a month revising it with Minkowski and the famous mathematician Hurwitz.
In August 1900, at the International Congress of Mathematicians in Paris, Hilbert delivered a famous lecture entitled "Mathematical Problems", in which Hilbert expounded the characteristics of the development of mathematics with his foresight, analyzed the role of internal and external factors in mathematics on mathematical progress, and emphasized that major mathematical problems are the guiding light for the progress of mathematics, the goose that will lay golden eggs, and can continuously produce new problems and new methods and new ideas. He was convinced that mathematics would not be divided into unrelated, isolated branches by the prevailing trend of specialization, and that the vitality of mathematics as a whole lay in the connection between its parts.
He also proposed 23 questions based on the results and trends of mathematical research in the 19th century, which are collectively known as the Hilbert problem. The Hilbert problem can be said to have become a beacon for the development of mathematics in the 20th century, guiding a direction for the future exploration of mathematics.
These 23 questions are shown in the following figure and belong to four major blocks:
Questions 1 to 6 are mathematical fundamentals; Problems 7 to 12 are number theory problems, and problems 13 to 1 8 belong to algebraic and geometric problems; Questions 1 9 through 2 3 belong to mathematical analysis. Sequentially, it is clear that Hilbert has focused his emphasis on mathematical foundations, and that his own work is working to create a solid foundation for the mathematical edifice. From the end of the 19th century Hilbert had worked to base mathematics on the axioms of the few. He was also one of the few earliest proponents of set theory, and it became his dream to base mathematics on set theory. This may explain why he listed the number one problem of set theory, the continuum hypothesis, as his first problem.
It can be said that the Hilbert problem involves most important areas of mathematics, which requires the author to have a rich and broad mathematical knowledge to be able to do it, it can be said that few mathematicians can see the problem so comprehensively and thoroughly, which is very familiar to everyone, it should be the problem of the distribution of prime numbers, especially for the Riemann conjecture, goldbach conjecture and twin prime number problems.
Prime numbers are a very old field of study. Hilbert mentioned the Riemann conjecture, the Goldbach conjecture, and the twin prime problem at the congress. The Riemann conjecture remains unresolved. The Goldbach conjecture and twin prime problems are also not finally solved, and the solution of these problems is of great significance for the study of prime numbers.
Among them, the best result of the Goldbach conjecture belongs to the Chinese mathematician Chen Jingrun (1+2), while the Chinese mathematician Zhang Yitang made a breakthrough contribution in the field of twin prime conjecture in 2013.
In his lecture, Hilbert proposed what he considered to be the perfect mathematical problem: the problem can be expressed concisely and clearly, but the solution is so difficult that a new way of thinking is necessary to achieve it.
To illustrate his point, Hilbert gives two most typical examples: the first is Fermat's Great Theorem, where the algebraic equation x^n + y^n = z^n has no non-zero integer solution when n is greater than 2; the second is the special case of the N-body problem——— the "three-body problem". It is worth mentioning that although these two issues had not been resolved at the time, Hilbert did not put them on his list of problems.
Wiles and Fermat's Last Theorem
Because of Hilbert's own great influence in the mathematical community, Hilbert 23 questions immediately became the goal that countless mathematical communities wanted to solve, and to this day, the Hilbert problem is still a difficult problem that many mathematicians try to overcome, which has had a profound impact on the research and development of modern mathematics and played a positive role in promoting it. It can be said that Hilbert, through his own work and the questions he posed, took 20th-century mathematics on a healthy path.
Hilbert's 23 questions are more of a continuation of the past, saying that it continues to summarize almost all the important problems that have not been solved in the 19th century; it is said that it is these problems that have indeed promoted the progress of mathematics in the 20th century. Therefore, mathematicians in the major mathematical countries, the United States, Russia, Japan, France, Germany and the United Kingdom, have organized or studied the history and current situation of the Hilbert problem alone, and have further asked new questions, and our national mathematical community has also made a lot of contributions to Hilbert 23.
Some of Hilbert's problems have now been satisfactorily resolved, and some remain unresolved. The belief that every mathematical problem can be solved in his speech is a great encouragement to mathematicians, and it also motivates mathematicians to explore the unknown fields of mathematics and explore the frontiers of mathematics in the future.
The great mathematician Weil once vividly commented on Hilbert and his 23 questions as followed: "Hilbert is like a bagpiper in variegated clothes, and the sweet flute has seduced many rats and jumped into the abyss of mathematics with him."